Chapter 3
Polynomial Fucntion
Factoring Sum or Difference of Cubes
Ex: 27x^3+125=(3x+5)(9x^2-15x+25)
a^3-b^3 = (a-b)(a^2+ab+b^2)
a^3+b^3 = (a+b)(a^2-ab+b^2)
Factoring Polynomial
Lastly, divide f(x) by (x-a)
Try each of them until f(a)=0
If f(a)=0, then (x-a) is a factor of f(x)
Possible values for "a" are the factors of the constant term (term with no variable) in f(x)
Ex: factors of 9 are +-1, +-2, +-3
To factor f(x) we need to find (x-a) where f(a)=0
The Remainder Theorem
Ex: divide f(x)=x^2-4 by x-2
f(2)=(2)^2-2=0
Therefore, (x-2) is a factor of f(x) meaning remainder is 0
When a polynomial f(x) is divided by (x-a), the remainder is equal to f(a)
If the remainder is equal to zero, then (x-a) is a factor of f(x)
Dividing Polynomial
Division Statement: Divisor * Quotient + Remainder
Synthetic Division
7. The last number(10) on the bottom row is the remainder and the rest are the co-efficient of the quotient.
6. Multiply the last number below the line by k and repeat step 3-5
5. Multiply the number below the line by k. Put the product above the line in the second column and then add it with the co-efficient above it
3*4=12
-2+12=10
4. Drop the first co-efficient
3. Copy the co-efficient inside the chart and place k or k/a to the left
1./2. Repeat step 1 and 2 from Long Division
Synthetic division can only be used when the divisor is linear Ex: (x-k)
Long Division
6. Repeat steps 3-5
5. Subtract the dividend with the product
4. Multiply the last term of quotient by the divisor, write the result below the dividend
3x*(2x+5)
3. Divide the first term of the dividend by the first term of the divisor
6x^2/2x=3x
2. Fill any missing terms from the general form of polynomial with zeroes
1. Write both the divisor and dividend in descending powers of the variables
End Behavior
For EVEN degree function
If leading co-efficient is negative, graph extends from 3rd quadrant to 4th quadrant
x → +-∞, y → -∞
If leading co-efficient is positive, graph extends from 2nd quadrant to 1st quadrant
x → +-∞, y → ∞
For ODD degree function
If leading co-efficient is negative, graph extends from 2nd quadrant to 4th quadrant
x → -∞, y → +∞
x → +∞, y → -∞
If leading co-efficient is positive, graph extends from 3rd quadrant to 1st quadrant.
x → -∞, y → -∞
x → +∞, y → +∞
Characteristic of Function
Leading Co-Efficient
is the co-efficient of the highest power
Symmetry
Most polynomial functions are neither
Odd function has symmetry over both the x and y-axis
Even function has symmetry over the y-axis
Order in Factored form
Order 3 is cubed(cubic shape)
Order 2 is quadratic(parabolic)
Order 1 is linear(curve crosses through x-axis)
Order of an x-intercept is its exponent
Ex: (x-3)^2 meaning x=3 has the order of 2
In factored form, we can find x-intercept
Roots
Function with even degree may have no roots
Function with odd degree must have at least 1 root because it has opposite end bahavior
Arms and Turning Points
Number of arms of a function is determined by its degree(most of the time)
An arm is a continuous lines in between 2 turning points
Number of turning points of a function is at most degree-1
Turning Points is the where the function change vertical direction
Degree
5th degree is quintic
4th degree is quartic
3rd degree is cubic
2nd degree is quadratic
1st degree is linear
is the exponent of the highest power
